Staunton Chess Sets
Before Staunton sets (the sets you have probably used your whole life) the look of sets were visually disparate enough that even at top levels, playing on someone else's board was a sizable disadvantage. There were four main types of set in Europe and many more across Asia. Here's a few to give you the idea of the problem that they faced:
Rooks were variable in height, while pawns, bishops, queens and kings usually had the same basic shape as each other. Also the ornateness of the pieces made them expensive to create and the height of their centre of mass (on most sets) made them topple easily. In 1849 the game makers Jaques of London decided that it was time to standardise, but get people to adopt the new design they decided to get a celebrity endorsement (which appears to be the first celebrity endorsement).
The celebrity of choice was Howard Staunton, a thespian who just happened to be considered the best chess player in the world. These were shortly before the days of having an official world champion, but Staunton was a known name in the 1840s. Jaques of London got him to sign and number the first 500 sets and official endorse it to the papers whenever he won a match.
It worked and from the 1851 Championship onward the set of choice was the Staunton set. This led to a trickle down into club play and eventually to basically all commercial sets. Apart from the endorsement there were several advantages to the design which led to its adoption. Firstly the pieces were much squatter with wide bases that led to them being more stable. While blitz play was not popular at the time this is essential for modern play.
The Staunton Set
Secondly, apart from the knights most of the pieces could be produced almost entirely on a lathe with only small finishing touches down by hand. This made them cheap enough to be mass produced and they were hard to break unlike the thinner pieces of the past. One big innovation was the introduction of the slit in the bishop's mitre which made them instantly recognisable from pawns. The coronet on the queen was also made consistent.
However what really made it take off around Europe was its popularity with soldiers in wars. Cheap sets were sent in large numbers to the front lines such as in the Crimea War which started in 1853. Soldiers on both sides would play during the long campaigns and it helped the design spread from Western Europe back to the Near East.